I'm planning to conduct undergraduate research on local vertex antimagic total coloring. Although I've seen several research papers related to this topic, I still don't know the significance and difference between the coloring induced by graph labeling to the general proper vertex coloring in terms of its application to practical problems.
The concept of antimagic labeling was introduced by Hartsfield and Ringel in 1990. Antimagic labeling of a graph $G$ with $q$ edges is a bijection from the set of edges to the set of positive integers $\{1,2,...,q\}$ such that all the vertex weights are pairwise distinct, where the weight of a vertex is defined in the same way as for magic graphs.
In 2017, Arumugam et al. observed a connection between a vertex proper $k$-coloring and antimagic labeling of a graph.
A vertex proper $k$-coloring, given an integer $k$ is a map $c: V(G) \longrightarrow S$ for $|S|=k$, such that no two adjacent vertices share a common color. The chromatic number of the graph $G$ denoted by $\chi (G)$ is the smallest integer $k$ for which the graph has a proper vertex $k$-coloring.
They defined this connection as a new notion called local vertex antimagic labeling. A bijection $f: E(G) \longrightarrow \{1,2,3,...,|E(G)|\}$ is called local vertex antimagic labeling if for any two adjacent vertices $u$ and $v$, $w(u) \neq w(v)$, where $w(u) = \sum_{e \in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. This labeling induces a proper vertex coloring of $G$ where the vertex $u$ is assigned the color $w(u)$. The same notion was introduced by Bensmail et al. independently.
Putri et al. extended the notion of local vertex antimagic labeling by labeling the vertices and edges of a graph $G$ to establish vertex coloring. A bijection $f: V(G) \cup E(G) \longrightarrow \{1,2,3,...,|V(G)|+|E(G)|\}$ is called local vertex antimagic total coloring if for any two adjacent vertices $v_1$ and $v_2$. $w(v_1) \neq w(v_2)$, where for $v \in G, w(v) = \sum_{e \in E(v)}$ $f(e)+f(v)$, where $E(v)$ and $V(v)$ are respectively the set of edges incident to $v$ and the set of vertices adjacent to $v$. Analogous to the local vertex antimagic labeling, any local vertex antimagic total labeling induces a proper vertex coloring of $G$, local vertex antimagic total chromatic number denoted by $\chi_{lvat}(G)$, which is the minimum number of colors taken over all colorings induced by local vertex antimagic total labelings of $G$.
Combining the notion of general proper $k$-coloring and the antimagic graph labeling. Many variations arise such as Local antimagic total coloring, however, I would like to know why is it important or what will be the significant change if I were to use the graph coloring using the antimagic labeling than the general proper $k$-coloring. Thank you so much in advance.
What's the difference between the vertex proper k-coloring and coloring using antimagic labeling? Is there a significant difference in coloring using antimagic labeling in terms of its application on practical problems such as optimization problems etc? In general, I would like to know the specific significance of studying the coloring of graphs using antimagic labeling. Why do we need to combine or what is the importance of combining the two notions of vertex proper k-coloring and antimagic labeling?